# Area in real K3-surfaces

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Ilia Itenberg, Grigory Mikhalkin

**19/1/20**Published in : arXiv:2001.06871

For a real K3-surface X, one can introduce areas of connected components of the real point set RX of X using a holomorphic symplectic form of X. These areas are defined up to simultaneous multiplication by a real number, so the areas of different components can be compared. For any real K3-surface admitting a suitable polarization of degree 2g−2 (where g is a positive integer) and such that RX has one non-spherical component and at least g spherical components, we prove that the area of the non-spherical component is greater than the area of every spherical component. We also prove that, for any real K3-surface with a real component of genus at least two, the area of this component is greater than the area of every spherical real component.